Investment risk and return belong in the same conversation. A 12% gain sounds attractive, but does it include dividends? Did it happen in one year or over several? How much did the investment fluctuate along the way? Moreover, how much purchasing power remained after inflation, taxes, and fees? This guide shows you how to answer those questions. It also works through the exercises in the provided pages step by step.
The $ symbol denotes U.S. dollars, as it does in the original problems. Historical figures here are teaching examples, not forecasts for investments today. The U.S. Securities and Exchange Commission (SEC) explains that an asset can generate returns through price appreciation, interest, or dividends. It also encourages investors to consider risk, time horizon, costs, and diversification.
What Are Investment Risk and Return?
Return is an investment’s gain or loss over a specified period. Risk describes uncertainty about that outcome and the possibility of losing money or missing a financial goal. Investors generally seek a higher expected return when taking on more risk. However, more risk does not guarantee more profit. It means that the range of possible outcomes can be wider. See the SEC’s definition of risk.
For example, a stock may earn 20% one year and lose 15% the next. A short-term U.S. Treasury bill usually has less price uncertainty if held to maturity. Even so, its nominal return might not keep up with inflation and federal income tax. Therefore, the right investment depends on when you need the money and what you need it to do.
Dollar Return, Percentage Return, and Total Return
When you own a stock, changes in its price and any dividends both contribute to your return. Suppose you paid P₀ for one share, its ending price is P₁, and it paid dividends D during the period. Then:
Dollar return per share = P₁ − P₀ + D
Percentage return = (P₁ − P₀ + D) ÷ P₀ × 100
If you own N shares, multiply the dollar return per share by N. In addition, count dividends even if the share price does not change. FINRA discusses the difference between price gains and total return.
Imagine buying 100 shares at $37 each. One year later, each share is worth $41 and paid $0.28 in dividends. Your initial investment was $3,700, the dividends totaled $28, and the shares gained $400 in value. Thus, your total dollar return was $428, or 11.57% of the initial investment. If you keep the shares, the $400 price gain remains unrealized. Nevertheless, measuring your economic return using the current market price gives the same result on that date. Your actual taxes and transaction costs may differ depending on the account and whether you sell.
Nominal Return Versus Real Return
A nominal return measures growth in currency units. A real return adjusts that growth for inflation. If the nominal return is r and inflation over the same period is π, calculate:
Real return = [(1 + r) ÷ (1 + π)] − 1
For example, an 8% nominal gain with 5% inflation produces a real return of about 2.86%: 1.08 ÷ 1.05 − 1. Subtracting 5 from 8 gives a useful approximation of 3%, but the formula is more precise. The Bureau of Labor Statistics explains how the Consumer Price Index can measure changes in purchasing power.
How Do You Calculate Average Returns Over Several Years?
The word “average” can answer different questions. Therefore, choose a measure that fits your goal: estimating a single year’s return, describing compounded wealth growth, or accounting for deposits and withdrawals.
Arithmetic Mean: Average Annual Return
Add the annual returns and divide by the number of years:
Arithmetic mean = (r₁ + r₂ + … + rₙ) ÷ n
If an investment earns +20% and then −10%, its arithmetic mean is 5%. This figure summarizes an average individual period in the sample. However, it does not show exactly how much your money grew across both years. FINRA notes that a simple average can overstate the impression of cumulative performance.
Geometric Mean: Compound Growth
The geometric mean incorporates the sequence of gains and losses:
Geometric mean = [(1 + r₁) × (1 + r₂) × … × (1 + rₙ)]^(1/n) − 1
Enter returns as decimals: 20% becomes 0.20, while −10% becomes −0.10. Thus, $100 grows to $120 after the first year and ends at $108 after the second. The compounded annual rate is √(1.20 × 0.90) − 1 ≈ 3.92%, below the 5% arithmetic mean. In other words, the geometric mean answers: “What steady annual rate would have produced the same final balance?” The factors 1 + r must be nonnegative; a 100% loss reduces the balance to zero.
Why Does Volatility Reduce Compound Growth?
Losses can be harder to recover from than they initially seem. After a 50% loss, you need a 100% gain to get back to the starting value. Suppose annual returns are +50% and −50%. Their arithmetic mean is zero, yet $100 becomes $75: 100 × 1.50 × 0.50. Accordingly, the geometric mean is −13.40% per year in this example. The CFA Institute discusses the impact of volatility on compounded growth.
Money-Weighted Returns: Deposits and Withdrawals
If you add more money immediately before strong performance, your personal return may exceed the strategy’s return. Conversely, a large deposit before a downturn can reduce your personal result. The internal rate of return (IRR) accounts for the size and timing of each cash flow. It is also called a money-weighted return.
By contrast, a time-weighted return links the returns between external cash flows. It is especially useful when evaluating a manager who does not control when a client deposits or withdraws money. The CFA Institute distinguishes the two methods in its explanation of the GIPS standards.
How Do Variance and Standard Deviation Measure Risk?
Standard deviation measures how far returns vary around their average. A higher figure means the observed returns were more dispersed. However, it does not reveal the exact probability of a loss, the depth of a future crash, or the risk of running short of money on a particular date.
For a sample of n returns, first calculate their average r̄. Next, subtract the average from each return, square each difference, and add the squared differences:
Sample variance = Σ(rᵢ − r̄)² ÷ (n − 1)
Sample standard deviation = √sample variance
The n − 1 denominator follows the sample convention used in the photographed exercises. If the observed years are the entire population you want to describe, population variance uses n instead. Keep the convention consistent when comparing investments. Also, if you enter returns as percentage figures, variance has units of squared percentage points. Standard deviation then has units of percentage points.
Normal Distribution: What Can It Tell You?
An idealized normal distribution is symmetric. Its standardized form has a mean of zero and a standard deviation of one. About 68% of observations fall within one standard deviation of the mean, while roughly 95% fall within two. To standardize an observation x, use z = (x − μ) ÷ σ.
For example, if the mean return is 12% and the standard deviation is 10%, a 2% return has a z-score of −1. Under a normal model, the chance of a return below 2% is about 15.87%. However, normality is a simplifying assumption. The CFA Institute explains that financial returns often have heavier tails than the normal distribution. Therefore, do not treat a normal-model probability as a reliable forecast for an actual stock without examining the assumptions.
Diversification and the Risk Premium
A risk premium compares a risky investment’s expected return with a lower-risk reference rate. The comparison should use the same currency and time horizon. For some examples, a short-term Treasury bill can serve as the reference:
Expected risk premium = expected asset return − reference rate
If a stock has an 11% expected return and the reference rate is 4%, its expected risk premium is 7 percentage points. That does not mean the investor will actually receive seven extra points. Moreover, a portfolio’s risk depends on how its assets move together, not only on each asset’s standard deviation. The SEC explains that diversification may reduce risk without eliminating losses.
How Do Inflation, Taxes, and Costs Change Your Return?
First, fees and expenses reduce the amount of money that remains invested. As a result, even small annual differences can compound over time. The SEC advises investors to check whether performance figures include costs.
Next, consider the relevant tax rules. For U.S. Treasury bills, TreasuryDirect states that interest is subject to federal income tax but exempt from state and local income taxes. Suppose, strictly as a classroom example, that a taxable 5% return faces an effective 40% federal tax on the interest. The after-tax nominal return would be 3%: 5% × (1 − 0.40). If inflation were also 3%, the after-tax real return would be zero: 1.03 ÷ 1.03 − 1. Actual tax treatment depends on your circumstances and the asset.
Finally, compare returns over the same length of time. A constant 1.63% monthly return compounded over 12 months equals (1.0163)^12 − 1 = 21.41% annually. Simply multiplying 1.63% by 12 gives 19.56% and misses compounding.
Solved Exercises: Return and Risk Calculations
The following problems paraphrase the questions while retaining the figures needed for each calculation.
Review Problem 1: Buying 400 Shares and Receiving Dividends
An investor buys 400 shares at $30 each, receives $0.75 per share in dividends, and ends the year with a share price of $33. The capital gain equals 400 × (33 − 30) = $1,200. In addition, dividends total 400 × 0.75 = $300. Thus, the total dollar return is $1,500. The initial investment was 400 × 30 = $12,000, so the percentage return is 1,500 ÷ 12,000 = 12.5%.
Review Problem 2: Two Stocks, Average Returns, and Volatility
The annual return series are Michele: 12%, −4%, 0%, 20%, 2% and Janicek: 5%, −15%, 10%, 38%, 17%. Here, variance follows the sample convention and uses a denominator of four.
| Measure | Michele | Janicek |
|---|---|---|
| Sum of five returns | 30% | 55% |
| Arithmetic mean | 6.00% | 11.00% |
| Sum of squared deviations | 384 | 1,478 |
| Sample variance | 96 | 369.5 |
| Sample standard deviation | 9.80% | 19.22% |
| Annual geometric mean | 5.65% | 9.65% |
For Michele, the squared deviations add up to 36 + 100 + 36 + 196 + 16 = 384, so sample variance is 384 ÷ 4 = 96. For Janicek, the sum is 36 + 676 + 1 + 729 + 36 = 1,478, giving 1,478 ÷ 4 = 369.5. Next, take each variance’s square root to find the standard deviation. Michele’s geometric mean is [(1.12)(0.96)(1.00)(1.20)(1.02)]^(1/5) − 1. Janicek’s is [(1.05)(0.85)(1.10)(1.38)(1.17)]^(1/5) − 1. In this sample, Janicek had a higher average return and greater dispersion.
Review Problem 3: Blume’s Formula for Different Horizons
A 30-year history has a 12.8% arithmetic mean and a 10.7% geometric mean. The textbook applies the following interpolation for a horizon T, given N = 30:
Blume estimate(T) = [(N − T) ÷ (N − 1)] × arithmetic mean + [(T − 1) ÷ (N − 1)] × geometric mean.
The estimate is 12.51% for 5 years, 12.15% for 10 years, and 11.42% for 20 years. For example, over five years: (25/29 × 12.8%) + (4/29 × 10.7%) = 12.51%. The formula blends the two averages according to the forecast horizon; it does not promise any return. See Marshall Blume’s original paper on estimating long-run rates of return.
Answer Key: Multiple-Choice Questions
Questions 1 Through 5
1. Maximum purchase price for a 15% return: You would receive $1.50 in dividends and sell the share for $26 after one year. Therefore, P₀ = (26 + 1.50) ÷ 1.15 = $23.91. Choice b.
2. Portfolio value after seven years: From January 1, 2004, through December 31, 2010, there are seven annual holding periods. At a 5% geometric mean, $100,000 × 1.05⁷ = $140,710. Choice b. The 6% arithmetic mean does not replace the compounded rate here.
3. Correct statement about standard deviation: It uses the same units as the original data. Choice c. Standard deviation is the square root of variance and cannot be negative.
4. Probability of earning less than 2%: z = (2 − 12) ÷ 10 = −1. For a normal distribution, the area below −1 is approximately 16%. Choice b.
5. Standard normal distribution: It has a mean of 0 and a standard deviation of 1. Choice b.
Questions 6 Through 10
6. Range containing about 95% of observations: With a mean of 100 and a standard deviation of 10, two standard deviations on either side give 80 to 120. Choice c.
7. Stocks, bonds, options, and futures: They are all financial assets. Choice d. They are not all debt instruments or ownership interests.
8. An investment doubles in one year: Its economic return is (2V − V) ÷ V = 100%, even if the investor does not sell. Choice a. A sale can affect taxes, but it does not change the market-value gain measured at that point.
9. Asset class whose annual returns most closely tracked inflation in the chapter’s historical comparison: Short-term U.S. Treasury bills. Choice a. This does not mean they perfectly protect purchasing power. Their real return can be negative, especially after tax.
10. Asset class with the highest average historical risk premium in the textbook’s data: Small-company stocks. Choice d. The conclusion describes a historical sample. It does not predict that small stocks will outperform in the future.
Questions 11 Through 15
11. Arithmetic mean of −10%, 40%, 0%, and 20%: (−10 + 40 + 0 + 20) ÷ 4 = 12.5%. Choice d.
12. Chance of a negative return with a 20% mean and a 10% standard deviation: z = (0 − 20) ÷ 10 = −2. Under the normality assumption, the probability is about 2.28%. The closest answer is 2.5%, choice d.
13. Incorrect statement about the normal distribution: The probability of an outcome above the mean is 50%. However, choice d says the probability of a negative value is always one-half. That is false unless the normal distribution has a mean of zero. Choice d.
14. z-score of 200 when the mean is 500 and the standard deviation is 150: (200 − 500) ÷ 150 = −2.00. Choice a.
15. Least appropriate description of a normal distribution: It is continuous, not discrete. Choice b. Its bell-shaped curve is symmetric, approaches the horizontal axis in its tails, and theoretically spans negative to positive infinity.
Concept Questions: Explained Answers
How Do Asset Classes Rank by Historical Risk and Return?
Concept question 1. In the chapter’s simplified historical comparison, the ascending order of risk is Treasury bills → long-term government bonds → large-company stocks → small-company stocks. Broadly, average historical returns follow the same ranking. However, a historical pattern across asset classes does not guarantee that a riskier asset wins every year. FINRA describes the historical relationship between these broad asset groups.
How Can a Stock Earn 4% If Its Price Did Not Change?
Concept question 2. It may have paid dividends equal to 4% of its initial share price. Consequently, an investor can earn a positive total return even when the quoted price ends where it started. A stock split or other distribution may also require adjustments to a price series. Check exactly what the quoted return includes.
Can an Unleveraged Stock Position Lose More Than 100%?
Concept question 3. If you buy a share entirely with your own money and assume no additional obligations, you can lose at most 100% of the amount invested. The share price cannot fall below zero. Thus, a return below −100% is impossible for that position. A normal distribution applied mechanically to simple stock returns would allow such an outcome, exposing a limitation of that model. Leverage, short sales, and derivatives can have different loss limits.
Which Mean Should You Use to Evaluate Ten Years of Investing?
Concept question 4. The arithmetic mean describes the average observed annual return. The geometric mean describes your money’s compounded annual growth over all ten years. To answer “At what annual compounded rate did my capital grow?”, use the geometric mean. If you made deposits and withdrawals, also calculate a money-weighted return to evaluate your personal experience.
When Is Blume’s Formula Useful?
Concept question 5. It offers a historical estimate for a specific future horizon when you know a return series’ arithmetic and geometric means. For a shorter horizon within the sample length, the formula gives more weight to the arithmetic mean. As the horizon lengthens, the geometric mean receives more weight. Nevertheless, structural changes, costs, and sample uncertainty still matter.
Why Were Treasury Bill Yields High in Some Historical Periods?
Concept question 6. This question refers to a table and figures not included in the available pages. Therefore, the specific years sought by the textbook cannot be identified reliably from the material provided. In general, short-term nominal yields tend to rise when expected inflation and monetary policy rates rise. To identify the exact years, consult the missing figures. You can also compare them with the Treasury bill series from Federal Reserve Economic Data (FRED).
What Happens to Risk Premiums After Adjusting for Inflation?
Concept question 7. Inflation generally lowers each asset class’s real return. However, subtracting the same inflation rate from two nominal returns for the same period leaves their difference in percentage points approximately unchanged. With exact real-return adjustments, the premium in percentage points becomes (rᵃ − rᵇ) ÷ (1 + π). Accordingly, risk premiums do not necessarily fall by the same number of points as individual real returns.
How Do Taxes Affect Returns, Premiums, and Volatility?
Concept question 8. Taxes on gains generally reduce after-tax returns, but the change in the risk premium depends on each asset’s tax treatment. Under a simplified uniform proportional tax on every return, with no other complications, both differences in returns and standard deviations would shrink proportionally. In practice, realization dates, loss deductions, exemptions, and income categories matter. Therefore, no single reduction in volatility applies to every investor.
What Remains of a 5% Treasury Bill Return After a 40% Tax?
Concept question 9. Under the exercise’s assumptions, the investor keeps a 3% nominal after-tax return: 5% × 60%. If inflation is also 3%, the after-tax real return is zero. Higher inflation makes it negative. Consequently, low nominal credit risk does not ensure purchasing-power growth. Check TreasuryDirect’s tax information before applying a textbook scenario to your circumstances.
Should You Put Your Entire Portfolio Into Small Stocks?
Concept question 10. A high historical average does not justify putting 100% of your savings into one asset class. You also need to withstand losses, meet emergencies, and fund goals at specific dates. Thus, a portfolio may combine different assets and risk levels. Diversification cannot prevent every loss, but it reduces dependence on one outcome.
An Additional Return Calculation
The question at the bottom of page 37 describes 100 shares bought at $37 each. They pay a $0.28 dividend per share and finish at a market price of $41. As calculated earlier, the dollar return is $428, and the percentage return is 11.57%. Even without a sale, the market-value calculation still shows $428 in economic return at that date: $400 of unrealized appreciation and $28 in dividends received.
Case Study: Deposits, Withdrawals, and Annualized Returns
The available page gives three scenarios, each with a $100 beginning value. One includes a $20 deposit at the end of year two; another includes a $10 withdrawal at that point. However, Vega’s return series and the beginning of the case appear on an earlier page that was not supplied. Consequently, the precise numerical answers to questions 1 and 2 cannot be verified from these pages. Selecting an answer without the missing data would be a guess.
Nevertheless, we can explain the method. If the five annual returns are r₁ through r₅, the answer to question 1 is [(1+r₁)(1+r₂)(1+r₃)(1+r₄)(1+r₅)]^(1/5) − 1.
For question 2, find the annual rate i that balances the cash flows. With a $20 deposit at the end of year two, solve 100(1+i)⁵ + 20(1+i)³ = ending value. With a $10 withdrawal, solve 100(1+i)⁵ − 10(1+i)³ = ending value. The ending value depends on the missing return path. From the investor’s cash-flow perspective, the same equations can instead show the initial investment and deposit as outflows, and the withdrawal and ending value as inflows.
Question 3: A 1.63% average monthly geometric return annualizes to (1.0163)^12 − 1 ≈ 21.41%. Choice c. This compounds a constant monthly rate; it does not forecast next year’s actual return.
Question 4: To assess a portfolio manager without rewarding or penalizing client decisions about deposits and withdrawals, use a time-weighted return, which compounds the correctly measured subperiod returns. Among the options shown, this corresponds to choice b, “geometric.” In practice, simply taking the geometric mean of annual returns may miss cash flows occurring within a year.
Exercises Using Online Price Data
The final page also asks the reader to look up ticker symbols and calculate measures for a chosen stock. The result depends on which company and historical window you select. First, record the ticker and retrieval date. Next, obtain monthly prices adjusted for corporate actions and calculate monthly returns as Pₜ/Pₜ₋₁ − 1. Then calculate the arithmetic mean, sample standard deviation, and geometric mean.
If you want a total-return measure, use a series that incorporates dividends; prices adjusted only for stock splits may not be sufficient. Furthermore, check whether the companies and tickers in an older textbook still trade under the same names.
Frequently Asked Questions About Investment Risk and Return
Does a High Standard Deviation Mean I Will Lose Money?
No. It describes the dispersion of returns, including gains as well as losses. Still, greater dispersion can complicate plans to withdraw money soon. Examine drawdowns, liquidity, fees, and the possibility of a permanent loss too.
Can the Arithmetic Mean Be Higher Than the Geometric Mean?
Yes. For a finite series of returns with positive growth factors, the geometric mean cannot exceed the arithmetic mean. As returns become more variable, the gap tends to widen. Therefore, do not automatically substitute a simple average when describing compounded wealth growth.
Does a 10% Return Protect Against Inflation?
It depends on inflation over the same period and what remains after fees and taxes. For example, with 12% inflation, a 10% nominal gain equals 1.10 ÷ 1.12 − 1 ≈ −1.79% in real terms, before any other deductions.
Can Historical Returns Predict the Future?
Not with certainty. Past results help you compare scenarios and understand the range of observed outcomes. However, the SEC warns that past performance does not necessarily predict future results. Consider risk, expected return, your time horizon, and your ability to absorb losses together.
Conclusion
To understand investment risk and return, start with total return, including both price changes and income received. Next, choose the right average: arithmetic for individual periods, geometric for compounded growth, and money-weighted for the experience of someone making deposits and withdrawals. Finally, consider volatility, inflation, taxes, fees, and the date when you will need the money. The solved exercises show why the same return series can tell different stories depending on the question you ask.
Note on the provided pages: The numeric answers to the case study beginning before page 40 require the missing earlier return series. The exact years requested in concept question 6 also require the textbook’s missing table and figures. Every other visible exercise with enough information is solved above.
